The vectorspace category at a point of a tame concealed algebra (Q1183302)

The paper investigates the relationship between the tame concealed algebras and critical vector space categories, both of them are important in the representation theory of finite-dimensional algebras and partially ordered sets. Let \(A\) be a finite-dimensional algebra over an algebraically closed field. With each simple module \(E(x)\) one may associate the following full subcategory \(S_ A(x)\) of \(A\)-mod, the category of all finitely generated left \(A\)-modules: \[ S_ A(x):=\{X\in A\text{-ind}\mid\Hom_ A (P(x),X)\neq 0=\Hom_ A(P(x),\tau X) \text{ and } X\not\cong P(x)\}, \] where \(P(x)\) is the projective cover of \(E(x)\) and \(\tau\) is the Auslander-Reiten translation. Then one can form the vector space category \({\mathcal S}_ A(x):=(\text{add }S_ A(x),\Hom_ A(P(x),-))\). Now assume that the algebra \(A\) is tame concealed and connected. Then one can define a defect function \(\partial: S_ A(x)\to\mathbb{Z}\) by sending \(M\) to \(\langle h,\underline{dim} M\rangle/\dim \Hom_ A(P(x),M)\), here \(h\) is the minimal positive radical vector of the quadratic form corresponding to the Euler form \(\langle , \rangle\) of \(A\) introduced by C. M. Ringel. Moreover, \(S_ A(x)\) is finite and becomes a poset by defining \(X\leq Y\) if and only if there exists an \(f\in\text{Hom}_ A(Y,X)\) with \(\Hom_ A(P(x),f)\neq 0\). Let \(L_ A(x)\) (resp., \(C_ A(x)\), \(R_ A(x)\)) be the direct sum of all modules in \(S_ A(x)\) with \(\partial(M)=-h_ x\) (resp., \(-h_ x<\partial(M)<0\), \(\partial(M)=0\)). The main results are the following: Theorem 1: (1) \(L_ A(x)\oplus C_ A(x)\oplus P(x)\) is a tilting module and \(L_ A(x)\) is the unique preprojective tilting complement to \(C_ A(x)\oplus P(x)\). (2) \(C_ A(x)\oplus R_ A(x)\oplus P(x)\) is a tilting module. (3) \(C_ A(x)\) is a critical poset with respect to \(\leq\). Theorem 2. (1) There is a unique directed critical vector space category in \(S_ A(x)\): it is \((\text{add }C_ A(x),\Hom_ A(P(x),-))\) and of type \({\mathcal C}(h_ x)\). (2) \((\text{add}(C_ A(x)\oplus R_ A(x)),\Hom_ A(P(x),-))\) (resp., \((\text{add}(C_ A(x)\oplus L_ A(x)),\Hom_ A(P(x),-))\)) is a domestic tubular extension (resp., coextension) of the vector space category \((\text{add }C_ A(x),\Hom_ A(P(x),-))\).